Final answer:
To calculate the flow rate from the needle, you can use Poiseuille's law, which states that the flow rate is directly proportional to the pressure difference across the needle and the fourth power of the radius of the needle, and inversely proportional to the length and viscosity of the fluid. Using the given values, the pressure at the entrance of the needle is approximately 3831 N/m².
Step-by-step explanation:
To calculate the flow rate from the needle, we can use Poiseuille's law, which states that the flow rate is directly proportional to the pressure difference across the needle and the fourth power of the radius of the needle, and inversely proportional to the length and viscosity of the fluid:
Q = (P2 - P1) * π * r4 / (8 * η * L)
Where:
- Q is the flow rate in cm³/s
- P2 is the pressure at the entrance of the needle in N/m²
- P1 is the pressure in the vein in N/m²
- r is the radius of the needle in cm
- η is the viscosity of the fluid in Ns/m²
- L is the length of the needle in cm
Given that the inner diameter of the needle is 0.10 cm, the radius would be 0.05 cm. The length of the needle is 3.0 cm. Since the fluid is saline solution, we can consider its viscosity to be similar to that of water, which is approximately 0.001 Ns/m².
Let P2 be the pressure needed at the entrance of the needle to cause the flow rate. Assuming the pressure in the vein is 0 N/m², we have:
Q = P2 * π * (0.05 cm)4 / (8 * 0.001 Ns/m² * 3.0 cm)
Using algebra, we can solve for P2:
P2 = Q * 2 * 0.001 Ns/m² * 3.0 cm / (π * (0.05 cm)4)
Substituting the given flow rate of 100N, we get:
P2 = 100N/cm³ * 2 * 0.001 Ns/m² * 3.0 cm / (π * (0.05 cm)4)
After solving the equation, the pressure at the entrance of the needle is approximately 3831 N/m².